Eigenvalues and Eigenfunctions of Woods Saxon Potential in PT Symmetric Quantum Mechanics

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1 Eigevalues ad Eigefuctios of Woods Saxo Potetial i PT Symmetric Quatum Mechaics Ayşe Berkdemir a, Cüeyt Berkdemir a ad Ramaza Sever b* a Departmet of Physics, Faculty of Arts ad Scieces, Erciyes Uiversity, Kayseri, Turkey b Departmet of Physics, Middle East Techical Uiversity, Akara, Turkey Abstract Usig the Nikiforov Uvarov method, we obtaied the eigevalues ad eigefuctios of the Woods Saxo potetial with the egative eergy levels based o the mathematical approach. Accordig to the PT Symmetric quatum mechaics, we exactly solved the time idepedet Shcrödiger equatio for the same potetial. Results are obtaied for the s- states. PACS: w; 3.65.Db;.3.Gp; 3.65.Ge Keywords: Nikiforov-Uvarov method, PT- symmetry, No-Hermitia Potetial, Woods Saxo potetial * Correspodig author: Ramaza Sever, sever@metu.edu.tr 1. troductio the appearace of o-relativistic quatum mechaics, oe usually chooses a real

2 (Hermitia) potetial to derive the real eergy eigevalues of the correspodig time idepedet Schrödiger equatio [1]. About six years ago, Beder ad his co-workers ad later others have studied several complex potetials o the PT - symmetric quatum mechaics. They showed that the eergy eigevalues of the Schrödiger equatio are real [,3,4,5,6,7,8]. Afterwards o Hermitia Hamiltoias with real or complex spectra have bee studied by usig umerical ad aalytical techiques [9,1,11]. Various differet methods have bee adopted for the solutio of the above metioed potetial cases. Such as, oe of these methods which makes it possible to preset the theory of special fuctios by startig from a differetial equatio has bee developed by Nikiforov ad Uvarov Method(NU) [1]. This method is based o solvig the time idepedet Schrödiger equatio by reductio to a geeralized equatio of hypergeometric type. this work, the Schrödiger equatio is solved by usig the NU method to esure the eergy eigevalues ad eigefuctios of boud state for real ad complex form of the Woods Saxo potetial. t is selected for a shell model for describig metallic clusters i a successful way ad for lightig the cetral part of the iteractio eutro with oe heavy ucleus [13,14]. This paper is arraged as follows: Sec. we briefly review Nikiforov-Uvarov method. Sec. solutio of the Schrödiger equatio with Woods-Saxo potetial is itroduced first ad later with PT-symmetric ad o-pt-symmetric o-hermitia forms of the q-deformed forms are obtaied. Sec. V we discuss the results.. Nikiforov-Uvarov Method NU method is based o the solutio of a geeral secod order liear differetial equatio with special orthogoal fuctios. So a o-relativistic Schrödiger equatio ca be solved by usig this method. Thus, for a give real or complex potetial, the Schrödiger equatio i oe dimesio is reduced to a geeralized equatio of hypergeometric type with a appropriate s = s( x) coordiate trasformatio. Therefore it ca be writte i the followig form, τ () s σ () s ψ () s + ψ () s + ψ() s = σ σ () s (1) where σ () s ad σ () s are polyomials, at most secod-degree, ad τ () s is a first-degree polyomial. Hece Schrödiger equatio ad the Schrödiger like equatios ca be solved by meas of the special potetials or some quatum mechaical problems. By usig the trasformatio ψ () s = φ() s y() s () Eq.(1) reduces ito a hypergeometric type form σ () sy + τ() sy + λy= (3) where φ () s satisfies φ() s / φ() s = π() s / σ() s. ys () is the hypergeometric type fuctio whose polyomial solutios are give by Rodrigues relatio B d y () s = σ () s ρ() s, ρ() s ds (4) where B is a ormalizig costat ad the weight fuctio ρ must satisfy the coditio i Ref.[1] ( σρ) = τρ. (5) The fuctio π ad the parameter λ required for this method are defied as

3 σ τ σ τ π = ± σ + kσ (6) ad λ = k + π. (7) O the other had, i order to fid the value of k, the expressio uder the square root must be square of polyomial. Thus, a ew eigevalue equatio for the Schrödiger equatio becomes ( 1) λ = λ = τ σ, ( = 1,,,... ) (8) where τ () s = τ () s + π() s, (9) ad it must have a egative derivative. 3. Woods Saxo Potetial A basic problem i the uclear physics is the motio of the free electros which have a coclusive ifluece o the abudace of metallic clusters. These electros are movig i well defied orbitals, aroud the cetral ucleus ad i a mea field potetial which is produced by the positively charged ios ad the rest of the electros. the mea field potetial, the details of the potetial are described by free parameters such as depth, width ad the slope of the potetial, which have to be fitted to experimetal observatio. Therefore, a mea field potetial is always empirical ad its a example ca be give as the Woods Saxo potetial [15] V V() r =, (1) where V is the potetial depth, r R a 1+ e R is the width of the potetial ad a its diffuseess ad a is the surface thickess which is usually adjusted to the experimetal values of ioizatio eergies. Schrödiger equatio i the spherical coordiates is + V() r ψ() r = Eψ() r (11) m or the radial part of it takes the form m V R () r + E R() r + r 1 qe α = (1) + where the radial wave fuctio ψ ( r) is writte as ψ ( r) = R( r) / r ad the coversios of r R r, 1/ a α are doe by isertig a arbitrary real costat q withi the potetial. additio, here we assume that ψ ( r) = (1/ r) R( r) is bouded as r. r e α Now, we apply a trasformatio to s = to get a form that NU method applicable. Thus by itroducig the followig dimesioal parameters me mv ε ( E ) β α α ( β ) (13) which leads to the geeralized of hypergeometric type give by Eq.(1): d R() s 1 qs dr() s εqs + ( εq βqs ) + β ε Rs ( ) ds s(1 qs) ds s (1 qs) =. (14)

4 After the compariso of Eq.(14) with Eq.(1), we obtai the correspodig polyomials as τ () s = 1 qs, σ() s = s(1 qs), σ () s = εq s + ( εq βq) s+ β ε. (15) Substitutig these polyomials i Eq.(6), we achieve π fuctio as qs 1 π () s = ± ( q + 4εq 4kq) s + 4( βq εq+ k) s+ 4( ε β) (16) takig σ () s = 1 qs. The costat k is determied from the correspodig statemet i Ref. [1], i.e., k β q q ( ε β ) 1, = ±. Afterwards, these two values for each k are replaced i Eq.(16) ad the followig possible solutios for π ( s) are obtaied as ( ε β 1) qs ε β for k= βq+ q ε β, qs 1 π () s = ± ( ε β + 1 ) qs ε β for k= βq q ε β t is clearly see that the eergy eigevalues are foud with a compariso of Eq.(7) ad Eq.(8). Therefore, the polyomial τ ( s) i Eq.(9), for which its derivative has a egative value, is established by a suitable choice of the polyomial π ( s) for k = β q q ε β from Eq.(17): qs 1 π() s = ( ε β + 1) qs ε β, ( ) ( ) τ () s = 1+ ε β q 3+ ε β s, τ () s = q 3+ ε β. (19) After substitutig the polyomials π ( s) ad τ ( s) ad also k we get ad Thus the parameter ε takes the form ( ) (17) (18) λ = q β ε β 1, () ( ) λ = λ = q 3+ ε β + ( 1) q, (1) β + 1 β ε = + +. () ( + 1) Substitutig the values of ε ad β from Eq.(13) ad the trasformatio α 1a / i Eq(), oe ca immediately determie the eergy eigevalues as ma V + 1 ma V E = + +. (3) ma ( + 1) Here, the idex is o-egative itegers with > ad the above equatio idicates that we deal with a family of the stadard Woods Saxo potetial. Of course it is clear that by imposig appropriate chages i the parameters a ad V, the idex describes the quatizatio of the boud states ad the eergy spectrum. t is illustrated i Fig.1, the Woods Saxo potetial give by Eq.(1) ad some of the iitial eergy levels for differet q values are preseted by choosig V = 5 MeV, R = 58. fm ad a =. 65 fm. E Fig. shows that the eergy eigevalues as a fuctio the discrete level of the parameter a. for differet values We ow fid the correspodig eigefuctios. The polyomial solutio of the

5 hypergeometric fuctio ys ( ) deped o the determiatio of weight fuctio ρ ( s). Thus ρ () s i Eq.(5) is calculated as ( qs) s ε β ρ() s = 1, (4) ad substitutig i the Rodrigues relatio i Eq.(4), the polyomial ys ( ) is orgaized i the followig form 1 ε β d + 1 ε β y() s B( 1 qs) s + = ( 1 qs) s. (5) ds Choosig q = 1, the polyomial solutio of y ( s) is expressed i terms of Jacobi Polyomials, which is oe of the orthogoal polyomials with weight fuctio (1 ss ) ( 1 i the closed iterval [, 1], givig [costat] P ε β, ) (1 s ) [16]. By substitutig π ( s ) ad σ () s i the expressio φ() s / φ() s = π() s / σ() s ad the solvig the resultig differetial equatio, the other part of the wave fuctio i Eq.() is foud as () s (1 s) s ε φ = β, (6) with q = 1. Combiig the Jacobi polyomials ad φ ( s) i Eq.(), the s state wave fuctios are foud to be ε β ( ε β, 1) ε β R () s = C (1 s) s P (1 ) s, (7) where C is a ew ormalizatio costat determied by R () sds= 1. For =, 1, the uormalized wavefuctios i terms of hypergeometric polyomials are show i Fig PT symmetric ad o-hermitia Woods-Saxo case We are ow goig to cosider differet forms of the stadard Woods Saxo potetial, amely at least oe of the parameters is imagiary. For a special case, we take the potetial parameters i Eq.(1) as V V ad α iα. Such a potetial is called as PT Symmetric but o Hermitia ad its shape becomes 1+ qcos( αr) iqsi( αr) V() r = V. (8) 1+ q + qcos( α r) By substitutig this potetial ito Eq.(11) ad makig similar operatios i obtaiig Eq.(3), we get the eergy eigevalues from the compariso of λ values of E mv α ( + 1) =, m 4 α ( + 1) (8) λ = q 1+ β + i ε β, (3) ( ) λ = λ = q 3+ i ε β + ( 1) q. (31) A positive eergy spectra is obtaied if ad oly if < 1, sice the eergy α eigevalues of Woods Saxo potetial are egative. By choosig parameter α as purely imagiary, the eergy eigevalues obtaied for PT symmetric ad o Hermitia Woods- Saxo potetial are ot similar to Eq.(4). mv

6 3.. No-PT symmetric ad o-hermitia Woods-Saxo case Aother form of the potetial is obtaied by makig the selectios of V iv ad α iα. t takes the form qsi( αr) + i(1 + qcos( αr)) V() r = V (3) 1+ q + qcos( α r) ad called as o PT symmetric ad also o Hermitia. Accordig to Refs. [17,18], this type of the complex potetial is a pseudo Hermitia ad its basic properties are studied itesively. Substitutig Eq.(33) ito Eq.(11), we get the value of the π ( s) as ( ε iβ + i) qs ε iβ fork= iβq+ iq ε iβ, qs i π () s = ± ( ε iβ i ) qs ε iβ for k= iβq iq ε iβ ad after choosig the appropriate k ad π, we ca write τ as τ ( s) = (1+ i ε iβ ) qs(3+ i ε i β ). (34) Thus, the eergy eigevalues are reduced to the followig form mv α ( + 1) mv E = +, m 4 α( 1) i (1) + 4 α but it has real plus imagiary eergy spectra. Here, α, is a arbitrary real parameter ad i = 1. Whe we cosider the real part of eergy eigevalues a acceptable result is obtaied whe mv < 1 coditio. However, the eergy spectrum is ot see at the α imagiary part of eergy eigevalues, sice it is idepedet of. 4. Coclusios The exact solutio of the radial schrödiger equatio with the Woods-Saxo potetial for the s-states. Nikiforov Uvarov method is used i the calculatios after trasformig Schrödiger equatio ito the hypergeometric type. Schrödiger equatio is also solved for the complex potetial case. The eergy eigevalues obtaied for real case are compared with the oes for the complex case. Accordig to the complex quatum mechaics [19], the eigevalues of the coversio α iα are ot simultaeously eigestates of PT operator. f oe lets α iα i the Woods Saxo potetial, it is foud that the eergy levels of the potetial are positive i cotrary to expectatio. Whe α ad V parameters are purely complex, it is see that the umber of discrete levels for boud states is give oly by the real part of eergy eigevalues. Therefore, if all the parameters of potetial remai purely real, it is clear that all boud eergies E with represet a boud state eergy spectrum. (33)

7 Refereces [1] A. Khare, B.P. Madal, Phys. Lett. A 7 () 53. [] C. M. Beder ad S. Boettcher, Phys. Rev. Lett. 8 (1998) 543; C. M. Beder, S. Boettcher, ad P. N. Meiseger, J. Math. Phys. 4 (1999) 1; C. M. Beder, G. V. Due, ad P. N. Meiseger, Phys. Lett. A 5 (1999) 7. [3] C. M. Beder ad G. V. Due, J. Math. Phys. 4 (1999) 4616; F. M. Ferádez, R. Guardiola, J. Ros, ad M. Zojil, J. Phys. A: Math. Ge 31 (1998) 115. [4] H. Eǧrifes, D. Demirha ad F. Büyükkılıc, Physica Scripta,, 9(1999) ibid, 195(1999). [5] G. A. Mezicescu, J. Phys. A: Math. Ge 33 () 4911; E. Delabaere ad D. T. Trih, J. Phys. A: Math. Ge 33 () [6] M. Zojil ad M. Tater, J. Phys. A 34 (1) 1793; C. M. Beder, G. V. Due, P. N. Meiseger, ad M. Şimşek, Phys. Lett. A 81 (1) 311. [7] K. C. Shi, J. Math. Phys. 4 (1) 513; ad O the reality of the eigevalues for a class of PT -symmetric oscillators, math-ph/113; C. K. Modal, K. Maji, ad S. P. Bhattacharyya, Phys. Lett. A 91 (1) 3. [8] R. Kretschmer ad L. Szymaowski, The iterpretatio of quatum-mechaical models with o-hermitia Hamiltoias ad real spectra, quat-ph/1554, G. S. Japaridze, J. Phys. 35 () 179. [9] B. Bagchi, C. Quese, Phys. Lett. A 73 () 85. [1] Z. Ahmed, Phys. Lett. A 8 (1) 343. [11] Ö. Yesiltas, M. Simsek, R. Sever, C. Tezca, Physica Scripta, 67 (3) 47. [1] A.F. Nikiforov, V.B. Uvarov, Special Fuctios of Mathematical Physics (Birkhauser, Basel, 1988). [13] A. Bulgac ad C. Lewekopf, Phys. Rev. Lett. 71 (1993) 413. [14]. Hamamato, S. Lukyaov ad X. Z. Zhag, Nucl. Phys. A 683 (1) 55. [15] M. Brack, Rev. Mod. Phys. 65 (1993) 677. [16] G. Szego, Orthagoal Polyomials (America Mathematical Society, New York, 1939). [17] L. Solombrio, Weak pseudo-hermiticity ad atiliear commutat, J. Math. Phys. 43, () [18] A. Mostafazadeh, J. Math. Phys., 43 () 5; ibid, 43 () 3944; ibid, 43 () 814. [19] C. M. Beder, D. C. Brody ad H. F. Joes, Complexs extetio of quatum mechaics, Physical Review Letters 89, () 74, C. M. Beder, Symmetry i Noliear Mathematical Physics, Proceedigs of the stitute of Mathematics of NAS of Ukraie, Vol. 5, part, (4)

8 Figure Captios Figure 1: Variatio of the Woods-Saxo potetial as a fuctio of r. The parameters take values V = 5 MeV, R = 58. fm ad a =. 68 fm. q is a arbitrary parameter. Figure : The variatio of the eergy eigevalues with respect to the quatum umber with V = 5 MeV. The curves correspod to the differet values of the rage parameter a. Figure 3: Variatio of the uormalized wave fuctios agaist the expoetial rage parameter s for the first three s-states defied i terms of hypergeometric polyomials.

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